On λ-invariants attached to cyclic cubic number fields

Abstract

We describe an algorithm for finding the coefficients of F(X) modulo powers of p, where p ≠2 is a prime number and F(X) is the power series associated to the zeta function of Kubota and Leopoldt. We next calculate the 5-adic and 7-adic λ-invariants attached to those cubic extensions K/Q with cyclic Galois group A₃ (up to field discriminant <10⁷), and also tabulate the class number of K(e2πi/p) for p=5 and p=7. If the λ-invariant is greater than zero, we then determine all the zeros for the corresponding branches of the p-adic L-function and deduce Λ-monogeneity for the class group tower over the cyclotomic Zp-extension of K.